On the existence of abelian surfaces with everywhere good reduction

Abstract

Let D ≤ 2000 D \le 2000 be a positive discriminant such that F = Q ( D ) F = \mathbf {Q}(\sqrt {D}) has narrow class number one, and A / F A/F an abelian surface of GL 2 \operatorname {GL}_2 -type with everywhere good reduction. Assuming that A A is modular, we show that A A is either a Q \mathbf {Q} -surface or is a base change from Q \mathbf {Q} of an abelian surface B B such that End Q ⁡ ( B ) = Z \operatorname {End}_\mathbf {Q}(B) = \mathbf {Z} , except for D = 353 , 421 , 1321 , 1597 D = 353, 421, 1321, 1597 and 1997 1997 . In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over F F for D = 353 , 421 D = 353, 421 and 1597 1597 , which are non-isogenous to their Galois conjugates. These are the first known such examples.